Serving Time. According to the Bureau of Crime Statistics and Research of Australia, as reported on Lawlink, the mean length of imprisonment for motor-vehicle theft offenders in Australia is 16.7 months. One hundred randomly selected motor-vehicle-theft offenders in Sydney, Australia, had a mean length of imprisonment of 17.8 months. At the$5\%$ significance level, do the data provide sufficient evidence to conclude that the mean length of imprisonment for motor-vehicle theft offenders in Sydney differs from the national mean in Australia? Assume that the population standard deviation of the lengths of imprisonment for motor-vehicle-theft offenders in Sydney is 6.0 months.

• The Null Hypothesis states that the event will not take place. Unless it is rejected, the null hypothesis has no influence on the study's outcome.
• Its sign is H0, which is pronounced H-naught.
• The Null Hypothesis is the logical polar opposite of the Alternate Hypothesis.
• The alternative hypothesis is accepted once the null hypothesis is rejected. The symbol for it is H1.

Given information:

The standard deviation is 6,the mean is 17.8,the sample size is 100 and the level of significance is $0.05.$

Calculation:

The hypothesis are,

$H_0:\mu =16.7$

$H_a:\mu \ne 16.7$

The test statistics is,

$z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}$

Substitute the given values in above equation.

$z=\frac{17.8-16.7}{\frac{6}{\sqrt{100}}}$

$=1.83$

The critical value for level of significance of $0.05$is $\pm 1.96$.

The graph is shown below.

Thus, the null hypothesis is not rejected and it is not reasonable to believe that the mean length of imprisonment for motor vehicle theft offenders in Sydney differ from the national mean in Australia.